On Approximate Solutions of Linear and Nonlinear Singular Integral Equations

Abstract

Singular integral equation theory has broad applications to theoretical and practical investigations in mathematics, mathematical physics, hydrodynamic and elasticity theory. This fact motivated many researchers to work on this field and their studies have showed that finding approximate solutions of linear and nonlinear singular integral equations in Banach spaces provides many applications even if their definite solutions cannot be found or if there are difficulties in finding them. Thus, the central theme of the recent studies is to develop effective approximate solution methods for the linear and nonlinear singular integral equations in Banach spaces. This chapter has been devoted to investigating approximate solutions of linear and nonlinear singular integral equations in Banach spaces using technical methods such as collocation method, quadrature method, Newton–Kantorovich method, monotonic operators method, and fixed point theory depending on the type of the equations. We provide sufficient conditions for the convergence of these methods and investigate some properties.